Different...But the Same

There are two typical ways grades are set up in a course: total points
and percentage weighted.
These two seem radically different, but are really the same basic
thing viewed from two different perspectives.

Total points says that, for instance, there will be 1000 points during
the semester. So if you earn 900 points (or more), you'll get an A.
If you earn 800 to 899 points, you'll get a B. And so on...

Percentage weighting would instead say that there will be 100% available
during the semester. If you earn 90% of that (or more), you'll get an A.
If you earn between 80% and 90%, you'll get a B. And so on...

The tricky part is in looking at the total semester layout:

Total Points

Percentage Weighted

5 Chapter Tests

500

5 Chapter Tests

50%

Cumulative Final

200

Cumulative Final

20%

Weekly Quizzes

100

Weekly Quizzes

10%

In-class Activities

100

In-class Activities

10%

Homework

100

Homework

10%

Total

1000

Total

100%

Hmm... Actually, that wasn't that tricky, either. So where does
it get confusing?

Well, most teachers don't come up with an exact multiple of 10 for
their points. Instead, they just have whatever points were appropriate
for that semester's activities. So, the reality of it might look more
like this:

Total Points

Percentage Weighted

5 Chapter Tests

100+80+90+120+90 = 480

5 Chapter Tests

50%

Cumulative Final

192

Cumulative Final

20%

Weekly Quizzes

7+8+7+8+9+8+5+6 = 48

Weekly Quizzes

10%

In-class Activities

8+12+12+7+9 = 48

In-class Activities

10%

Homework

9+10+10+9+10 = 48

Homework

10%

Total

960

Total

100%

*shiver* Now the total points side looks quite scary. But it might also
look like this:

Total Points

Percentage Weighted

5 Chapter Tests

100+80+90+120+90=480

5 Chapter Tests

52%

Cumulative Final

250

Cumulative Final

27%

Weekly Quizzes

10+10+10+10+10+10+10+10=80

Weekly Quizzes

9%

In-class Activities

10+15+15+10+10=60

In-class Activities

7%

Homework

10+10+10+10+10=50

Homework

5%

Total

920

Total

100%

Of course those percentages aren't exact, but rounded to the nearest
percent.

Now the percentages look pretty scary, too.
Yet, the two sides are still [essentially] the same!

How Do I Know What I'm Getting Along The Way?

That's the real ickiness! Let's look at a couple of examples:

Total Points

(Reminder of semester schedule for total points:

Total Points

Assignment Item

Points It's Worth

Quiz 1

7

Quiz 2

8

Homework 1

9

Test 1

100

Quiz 3

7

Activity 1

8

Homework 2

10

Test 2

80

Assignment Item

Points It's Worth

Quiz 7

5

Activity 4

7

Homework 5

10

Test 5

90

Quiz 8

6

Activity 5

9

Final

192

Total

960

Assignment Item

Points It's Worth

Quiz 4

8

Quiz 5

9

Activity 2

12

Homework 3

10

Test 3

90

Quiz 6

8

Activity 3

12

Homework 4

9

Test 4

120

)

Let's take a typical student, say Phong. Let's take a typical class,
say Trigonometry (trig). As the semester progresses, Phong finds
that she's earned a 65 on the first test, gotten an 8 and a 0 on the
first two quizzes, and earned a 9 on the first chapter's homework.
How's Phong doing?

In an alternate universe, Stu is also taking trig. Stu also gets
a 65 (out of 100) on the first test. But he gets a 5 and a half
(out of 7) on the first quiz and a 0 (out of 8) on the second quiz.
And finally, he earns 8 (out of 9) on the first homework set.
How's Stu doing?

(Where'd those last numbers come from?! The 14.5 is just the total
from the second line: 10+1.25+1.25+2=14.5. But the 9.25 is a
little more weird. Stu got a 65 out of 100 on the first test, right?
That's 65%. But that 65% only counts for 10% of his semester grade
so it is really .65*10=6.5. So 6.5% of Stu's overall semester grade
has been earned from that first test. (Or, similarly, Stu has lost
3.5% of his overall grade to munging up
that first test.)

(Um...why did the 65% become .65 but the 10% stayed 10? Oh...er... We
can do it the right way, but the numbers just become horribly nasty:
.65*.1=.065. This still represents 6.5% of Stu's overall semester grade.
Just it is also in decimal. *shrug* Anyway, back to the 9.25...)

Then Stu got 5.5/7≅79% on the first quiz which is 1.25% of his
overall semester grade. So that represents .79*1.25≅1% of his
overall semester grade.

Next came the 0/8 on the second quiz. Well, 0 out of anything is
still 0% and 0% of anything is likewise 0%. So that quiz didn't do
much to improve Stu's semester grade...

Finally he earned 8 out of 9 which is approximately 89% on the first
homework set. That homework set counts for 2% of his overall grade,
so that means his score gives him another .89*2≅1.75% of his
semester grade.

Putting all of these pieces together: 6.5+1+0+1.75=9.25. So, Stu
has earned 9.25 of the 14.5 percent of the overall semester grade
which is approximately 63%.

Comparison

Whoa! How'd Stu end up in the same place as Phong?! It looked
like she was skunking him on the quizzes and homework... But
look at the breakdown side by side:

Phong

Stu

Task

Earned

Possible

Percent

Earned

Possible

Percent

Quiz 1

8

10

80

5.5

7

78.5

Quiz 2

0

10

0

0.0

8

0.0

Homework 1

9

10

90

8.0

9

89.0

Test 1

65

100

65

65.0

100

65.0

Now it is more obvious that Phong is only a slight bit ahead
on each item. So how did Stu stay even in the overall? Let's
add a column:

Phong

Stu

Task

Earned

Possible

Percent

Relative Worth

Earned

Possible

Percent

Relative Worth

Quiz 1

8

10

80

7.7%

5.5

7

78.5

8%

Quiz 2

0

10

0

7.7%

0

8

0

8%

Homework 1

9

10

90

7.7%

8

9

89

14%

Test 1

65

100

65

77%

65

100

65

69%

Wow! Things are worth a lot different portion of the grade than one
another (well, the test and homework, anyway).

Can I Still Pass?

The next most popular question is, "Can I still pass?" (Or similarly,
"What do I need to get a ____?")

Before we take a look, let's clear up a common misconception:

What Does Passing Mean?

Passing in college is almost exclusively a C (70%). A 'D' might have
meant 'Done' before, but now it is just a synonym for 'FaileD'.

Now back to our parallel universes:

Phong (Total Points)

To find out if Phong can pass, we need to know what passing means
for her:

Total Points: 920
Passing: 70% * 0.70
Points to pass: 644

Phong has 82 points and will need 644 to pass. That leaves:

Needed to Pass: 644
Phong has: - 82
Phong needs: 562

That means Phong will need to earn 562 of the remaining:

Total Points: 920
Points Done: - 130
Points Left: 790

790 points. She'll have to get, essentially:

Phong needs: 562
Points Left: 790
Needed %: 72%

72% (average) on all remaining material to pass the course. Since she
currently has a 63%, that could be do-able.
Looking at her scores
she seems to do fine on home out-of-class work and small in-class work.
Maybe she's just got test anxiety. A trip to Student Success
Services might be able to help!

Stu (Percentage Weighted)

Stu, on the other hand, has:

[Total Percent: 100]
Percent Done: - 14.5
Percent Left: 85.5

85.5% of his grade as yet undertermined. He's earned 9.25 of the 14.5
percent he's completed (see above). So in order for him to pass, he'll
have to get:

Passing: 70
Stu Has: - 9.25
Stu Needs: 60.75

60.75 of those 85.5 remaining percent. What's that? Well, that'd be:

Stu Needs: 60.75
Percent Left: 85.5
Needed %: 72%

*gasp* Stu needs 72% (average) on his remaining material to pass the
course as well! And, like Phong, since he currently has ~63%, that
seems like a reasonable goal. He even has the same kinds of problems
as she does and so might be amenable to similar help (i.e. a trip to
Student Success Services for a Test Anxiety Workshop or
Test Taking Strategies).

I Want to Do Better Than Just Pass!

Okay...what if Phong or Stu wants to get a B (80%) or even an A (90%)?

Phong (Total Points)

Stu (Percentage Weighted)

Pass (C; 70%)

(920*0.70-82)/(920-130) ≅ 72% (do-able)

(70-9.25)/(100-14.5) ≅ 72% (do-able)

B (80%)

(920*0.80-82)/(920-130) ≅ 83% (real hard)

(80-9.25)/(100-14.5) ≅ 83% (real hard)

A (90%)

(920*0.90-82)/(920-130) ≅ 95% (*snicker*)

(90-9.25)/(100-14.5) ≅ 95% (*snicker*)

So getting a C seems to be reasonable. But the B is going to
be real difficult to achieve. The A might be a little on the
'impossible' side given the current track record. These students
are going to have to really embrace the lessons given by Student
Success Services to make that kind of improvement.

Ongoing Grade Tracking

Let's see how things are going a little later, after our two students
have gotten their help and begin to perform a little better on those
tests:

How did they separate?! They still seem to be running neck-and-neck...
don't they? Let's see:

Phong

Stu

Task

Score (%)

Relative Weight to These

Relative Weight to Total so Far

Score (%)

Relative Weight to These

Relative Weight to Total so Far

Quiz 3

7/7=100%

7/105=7%

7/235=3%

15/15=100%

1.25/15.25=9%

1.25/29.5=4%

Activity 1

6.5/8=80%

8/105=8%

8/235=4%

16/20=80%

2/15.25=13%

2/29.5=7%

Homework 2

9.5/10=95%

10/105=9%

10/235=4%

9.5/10=95%

2/15.25=13%

2/29.5=7%

Test 2

57.5/80=72%

80/105=76%

80/235=34%

79.25/110=72%

10/15.25=65%

10/29.5=33%

Now we see that, although their individual scores are progressing
apace, the weights of their tasks to one another are not proportional
(across the universe divide, as it were).

The instructor in total-points world has a chapter 1 test that is worth
100 points of the total 920 (≅ 11%) whereas their chapter 2 test is
only worth 80 points of the 920 total (≅ 9%). Even though the
total points tests taken as a whole are [approximately] 52% of the overall
grade (480/920), they are not all equal to one another.

Over in percentage-weighted world, however, the instructor has allotted
50% of the overall grade to tests and since there are 5 chapter tests,
that comes out to 10% each. Note that the points on the test are
not relevant here! The chapter 1 test had 100 points
just like the one in Phong's world but Stu's chapter 2 test had 110
points which is more than the other world's 80. Both students also
scored approximately the same percentage on each test -- their performance
on each chapter is nearly identical to one another. But at this
stage, Phong's tests are not equivalent to one another whereas
Stu's are -- regardless of their point total!!!

By the time the semester is over, the two will work out close to the
same if they keep improving their performance at the same rate. However,
at different stages of the semester, Stu might be ahead of Phong (as now),
Phong might be ahead of Stu, or the two might be tied (as they were
before).

Moral: Comparing yourself to friends in other sections isn't necessarily
going to work out very well...

Continuing Chances At Success

Okay...what if Phong or Stu wants to: pass (C;
70%), get a B (80%), or even get an A (90%)?

Phong (Total Points)

Stu (Percentage Weighted)

Pass (C; 70%)

(920*0.70-162.5)/(920-235) ≅ 71% (fair)

(70-21.2)/(100-29.5) ≅ 70% (fair)

B (80%)

(920*0.80-162.5)/(920-235) ≅ 84% (tricky)

(80-21.2)/(100-29.5) ≅ 84% (tricky)

A (90%)

(920*0.90-162.5)/(920-235) ≅ 98% (yeow!)

(90-21.2)/(100-29.5) ≅ 98% (yeow!)

So getting a C seems like it won't be a stretch.
But the B is going to be fairly tricky to
achieve. The A seems to be getting even more
'impossible' than before.

The reason for this is that the students are incorporating those
Student Success Services lessons as best they can, but changing
long-standing negative behaviors is really difficult for most people. If
they'd perfectly learned how to Avoid Test Anxiety or employ
Improved Test Taking Strategies, they'd have gotten not 72% on
that chapter 2 test, but more like 88-96%. (This estimate is based on
looking at their other assignments for that chapter and realizing that an
in-class activity probably involved a group who may or may not have been of
equal caliber to our focal pupils.) Had that happened, their current
performance would have risen farther than ≅70% and they might still be
in reach of that A everyone truly desires. But
it just isn't going to happen for most people. Sometimes the reasons we
start a semester poorly are amenable to changes/fixes...sometimes they
aren't or are more difficult to change/fix.

Moral: Don't beat yourself up because something has happened, just
do the best with what you have. (You know, that whole lemons ⇒
lemonade thing...)

If these students take the lessons given by Student Success
Services to heart, they might make it out of trig not just alive, but
smelling like violets or petunias (if not roses).

One More Time!

Let's take a last look at our students after the chapter 3 test is
done to see how they are getting along:

See, now that Phong has another point-heavier test she's starting to catch
up... What about their prospectives for the future?

Phong (Total Points)

Stu (Percentage Weighted)

Pass (C; 70%)

(920*0.70-268.5)/(920-364) ≅ 68% (easy)

(70-34.85)/(100-46) ≅ 66% (easy)

B (80%)

(920*0.80-268.5)/(920-364) ≅ 85% (do-able)

(80-34.85)/(100-46) ≅ 84% (do-able)

A (90%)

(920*0.90-268.5)/(920-364) ≅ 101% (hey!)

(90-34.85)/(100-46) ≅ 103% (hey!)

So getting a C now looks like an easy ride --
just maintain what they are doing.

The B is still getting a little further away, but
it is now within reach of our grasp. Looking back over our recent track
record, our students are nearing the 80% mark on their tests and keeping
their A-B scores on
other portions of their grade. Their test scores are going up about 7-8%
each time so they should be there by the next test (and they still have two
chapters left so that is definitely a possibility.)

That A is now scooting beyond nasty. In fact
it's getting harder for Stu than for Phong. Now they'd each have to not
only do everything pretty much perfectly, but even hand in a bunch of extra
credit! *sigh* Who's got time to do that with other classes and family
issues and working to pay for school, insurance, gas, car payment, etc.
*shakes head*

*shrug* At least that B came down to Earth.
*bounce* (Again, lemons ⇒ lemonade! Don't look at the goal you can't
reach without minor mental/physical trauma! Look at the awesome goal just
over that ridge. Isn't it beautiful? A B for
someone who flunked the first exam and missed a quiz is pretty good. And
they didn't even have to put in much extra effort -- just show up so they
didn't miss any more quizzes and get some tips on Studying and
Anxiety Management from the folks over at Student Success
Services...)